Question

Explain two - way ANOVA. What is the statistical information difference between this and a one way ANOVA.

Answer 1

Two-Way Classification: (or withrs one Observation percell: Let Us consider there are stwo Observations factors which may EHect the variable values 2 . suppose N observations are divided into h different classes & Each group containing K values ie, N=hxk. Hence the whole data can be expressed in hxk two-way table Total means Treatments o 22 dB--- AIK 22 023-t-dek, T2 Jan nel Blool an Total / T., mean T., die T.2 7.2 naa - Trik T.3--- T.K 1.3 ---Tok The total variation of any observation Split into 3 component the Variation due to i group (BLOCK) & The variation due to i Treatment The variation due to chance factor

Mathematical Model: The linear mathematical model z nig - Hij + Eij Assumptions: O Al Observations are at oure Indepen- dent 2 Different Effects are additive in nature 3) ei are id are loove) Hypothesis: the Hypothesis under these classification are HOB: There is significance difference among the Blocks that is x = K2 = – tk HOT: There is significance difference among the treatments that is Bi=B2=----Bk Statistical Analysis: let di- mean of th Block E j (i=1,2)---sh) aga mean of ith Treatment h t =1,2,---k) Za overall mean h saj = (*] pij etion

- The mathematical modei reduces to ni = 7+ (7. - 2- .) + (7.3 - 1.t (ris- ni. - neith- Transfering .. to the left side Squaring y summering both sides over i=1 to h, 3=1 tok we get THEY - ??(GT-ā-)=f(aj; -āsomhetā :) (nr.2..) + (ng - e.." - $ $(^; -ās -ājā.) + s (T; - ;) = ? (ł; - - +2 5 (017-ā;-.; -) (---) +251 (0:-5--) (-; -T.) +255 (ā;-7-) (Foso - Tag tä- Since Est..) (agen - Rozto.) = ( -2..) & (e-7; – 77 + 2). (3-1) (93-7:)-> (73-7--)} Wi W. Wir

i Sum of sovuqres: *$$( -ū..) EST is called total sum oL SQOQres Eelais-Ti-Toptan St is called A total som of Errors Elain. SB is called sum of Squares due to Blocks ex ( 2 1.-)" = St is called som of due to Treatments . The mathematical model is TSS-55€ +SSBUSST Degrest of freedom It is the size of Sample selected Independetty from the population from total no. of Observations one over all mean is substracted. Hence the df of Tss is hk-1 from total no: o observation ke Treatment means h block means are substracted E one over all mean is added Hence the af of sst is hk- k-hti (6-1) (k-1) from h block means one over an mean is substracted hence de of ssg is ni

from k Treatment means one over an mean is substracted Hence at a ss is n-1 From t Treatment means one overal mean is substracted Hence at of SST is K-1 Sum of Souares: * which obtained dividing the Sum of squares by af * Thus mean SUM OF Sovvares due to blocks, SB - SSB * Thus mean sum of squares due to Treatments &t=55T kol * Thus mean sum of Souares due to Errors, S - SS € (n-D (K-1) E- statistic By definition of F-static, for Blocks Treatment can be defined as Ft = St NF (k-1), (n-D(K-1) FB - BB OF (1-1) (6-1) (x-1) se

df F-value ANOVA Table: source of sum mean 아 variation son Sou Qres Sovores Blocks h-) . SSB Treatments K-1 ssT Error Chock- sse. Total hk TSS SB se conclusion: It calculated value of Fis ress than table value of F at ry loss & corresponding then we accept * Otherwise we reject the hypothesis

1. m mi The basic differences blw one-way and two-way . The mathematical linear model in decomposed into - two components in one way and there components in two-way 7 In one-way classification only one factor is considered for analyazation. In two-way classification two factors o - are considered sincultaneously, 3) The degrees of freedom for error sum of suquares is 1. less than in two-way classification compared to one- OVO LIN way classification. y classification compared to one- CScanned with CamScanner